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7 PRIMITIVES AND INTEGRALS 165

(b) Show that at every point t e I — D where // is continuous on the left, / has a
derivative on the left. If at / e I — D,/i' is continuous,/has a derivative at the point /.
(Use (a).)

3. Let / be a differentiate mapping of an open subset A of E into F (E, F Banach
spaces).

(a) In order that /' be continuous at x₀, a necessary and sufficient condition is
that, for any e > 0, there exist 8 > 0 such that the relations \\s\\ =^ 8, ||/|| ^ S imply
||/Oo +s) -f(x₀ +1) ~f'(x₀) • (s - r)|i ^ e\\s - t\\.

(b) In order that /' be uniformly continuous in A, a necessary condition is that, for
any e>0, there exist 8 >0 such that the relations \\s\\ ^ 8, x e A, x -j-fy e A for
0 ^ £ < 1 imply \\f(x + s) -f(x) —f'(x) -s\\^ e\\s\\. The condition is sufficient when
A is convex (Section 8.5, Problem 8).

4. Let/be a continuous mapping of a compact interval I c R into R, having a continuous
derivative in I. Let S be the set of points /el such that /'(/) = 0. Show that for any

€ > 0, there exist a sequence (r_n) of numbers > 0 such that ^ r_n ^ e and that the set

n = 0

/(S) is contained in a denumerable union of intervals J_n, such that S(J_n) =^ r_n. (For any
a > 0, consider the open subset U_a of I consisting of the points t where |/'(OI < ^
use (3.19.6) and the mean value theorem.)

5. Let /be a continuous mapping of an interval I ^c R into C, such that /(/) ^ 0 in I and
and that //(/) exists in the complement of a denumerable subset D of I. In order that
|/| be an increasing function in I, show that a necessary and sufficient condition is that
#C/i'(0//(0) ^ 0 in I - D.

6. Let E, F be two Banach spaces, A an open subset of E, B a closed subset of the sub-
space A, whose interior is empty and such that any segment in E which is not contained
in B has an at most denumerable intersection with B. Let/be a continuously differen-
tiable mapping of A — B into F, and suppose that at each point b e B, the limit of
f'(x) with respect to A — B exists. Show that/can be extended by continuity to a con-
tinuously differentiate mapping/ of A into F (same method as in Problem 2(a)).

7. PRIMITIVES AND INTEGRALS

Let / be a mapping of an interval I cr R into a Banach space F. We
say that a continuous mapping g of I into F is a primitive of/in I if there
exists a denumerable set D c I such that, for any { e I — D, g is differentiate

at£ and </(£) =/(£)•

(8.7.1) Ifg^ g₂ are two primitives of f in I, then g_i — g₂ is constant in I.
This follows at once from the remark following (8.6.1).

Any interval I in R (not reduced to a point) is the union of an increasing
sequence of compact intervals J_n; to check that a function / defined in I
has a primitive, it is only necessary to do so for the restriction of/to each



	7 PRIMITIVES AND INTEGRALS 165 (b) Show that at every point t e I — D where // is continuous on the left, / has a derivative on the left. If at / e I — D,/i' is continuous,/has a derivative at the point /. (Use (a).) 3. Let / be a differentiate mapping of an open subset A of E into F (E, F Banach spaces). (a) In order that /' be continuous at x₀, a necessary and sufficient condition is that, for any e > 0, there exist 8 > 0 such that the relations \\s\\ =^ 8, \|\|/\|\| ^ S imply \|\|/Oo +s) -f(x₀ +1) ~f'(x₀) • (s - r)\|i ^ e\\s - t\\. (b) In order that /' be uniformly continuous in A, a necessary condition is that, for any e>0, there exist 8 >0 such that the relations \\s\\ ^ 8, x e A, x -j-fy e A for 0 ^ £ < 1 imply \\f(x + s) -f(x) —f'(x) -s\\^ e\\s\\. The condition is sufficient when A is convex (Section 8.5, Problem 8). 4. Let/be a continuous mapping of a compact interval I c R into R, having a continuous derivative in I. Let S be the set of points /el such that /'(/) = 0. Show that for any 00 € > 0, there exist a sequence (r_n) of numbers > 0 such that ^ r_n ^ e and that the set n = 0 /(S) is contained in a denumerable union of intervals J_n, such that S(J_n) =^ r_n. (For any a > 0, consider the open subset U_a of I consisting of the points t where \|/'(OI < ^ use (3.19.6) and the mean value theorem.) 5. Let /be a continuous mapping of an interval I ^c R into C, such that /(/) ^ 0 in I and and that //(/) exists in the complement of a denumerable subset D of I. In order that \|/\| be an increasing function in I, show that a necessary and sufficient condition is that #C/i'(0//(0) ^ 0 in I - D. 6. Let E, F be two Banach spaces, A an open subset of E, B a closed subset of the sub- space A, whose interior is empty and such that any segment in E which is not contained in B has an at most denumerable intersection with B. Let/be a continuously differen- tiable mapping of A — B into F, and suppose that at each point b e B, the limit of f'(x) with respect to A — B exists. Show that/can be extended by continuity to a con- tinuously differentiate mapping/ of A into F (same method as in Problem 2(a)).

	7. PRIMITIVES AND INTEGRALS Let / be a mapping of an interval I cr R into a Banach space F. We say that a continuous mapping g of I into F is a primitive of/in I if there exists a denumerable set D c I such that, for any { e I — D, g is differentiate at£ and </(£) =/(£)•

	(8.7.1) Ifg^ g₂ are two primitives of f in I, then g_i — g₂ is constant in I. This follows at once from the remark following (8.6.1). Any interval I in R (not reduced to a point) is the union of an increasing sequence of compact intervals J_n; to check that a function / defined in I has a primitive, it is only necessary to do so for the restriction of/to each